CIMNE Seminar - "hp-Adaptivity for Dummies", by David Pardo

Published: 04/02/2019

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Tuesday, February 19th, 2019. Time: 12 p.m.

Place: O.C. Zienkiewicz Conference Room, C1 Building, UPC Campus Nord, Barcelona


Despite the existence of several hp-adaptive algorithms in the literature (e.g. [1]), very few are used in industrial context due to their high implementational complexity, computational cost, or both.

This occurs mainly because of two limitations associated with hp-adaptive methods:

A) The data structures needed to support hp-refined meshes are often complex.

B) The design of a robust automatic hp-adaptive strategy is challenging.
To overcome limitation A, we adopt the multi-level approach of D'Angela et al. [2]. This method handles hanging nodes via a multilevel technique with massive use of Dirichlet nodes.

Our main contribution in this work is intended to overcome limitation B by introducing a novel automatic hp-adaptive strategy. For that, we have developed a simple energy-based coarsening approach that takes advantage of the hierarchical structure of the basis functions. Given any grid, the main idea consists of detecting those unknowns that contribute least to the energy norm, and remove them. Once a sufficient level of unrefinement is achieved, a global h, p, or any other type of refinements can be performed.

We tested and analyzed our algorithm on one-dimensional (1D) and two-dimensional (2D) benchmark cases. The obtained adapted meshes are competitive with the optimal ones shown in [1].
In this presentation, we shall illustrate the main advantages and limitations of the proposed $hp$-adapted method.

[1] L. Demkowicz. Computing with hp-adaptive finite elements. Vol. 1. One and two dimensional elliptic and Maxwell problems. Applied Mathematics and Nonlinear Science Series. Chapman & Hall/CRC, Boca Raton, FL, 2007. ISBN 978-1-58488-671-6; 1-58488-671-4.

[2] D. D’Angella, N. Zander, S. Kollmannsberger, F. Frischmann, E. Rank, A. Schroder, and A. Reali. Multi-level hp-adaptivity and explicit error estimation. Advanced Modeling and Simulation in Engineering Sciences, 3(1):33, 2016. ISSN 2213-7467.


David Pardo Zubiaur is a Research Professor at Ikerbasque, the University of the Basque Country UPV/EHU, and the Basque Center for Applied Mathematics (BCAM). B.S. in Mathematics by the UPV (2000), Master (2003) & PhD (2004) in Computational and Applied Mathematics by the University of Texas at Austin. He has published over 160 research articles and he has given over 260 presentations. In 2011, he was awarded as the best Spanish young researcher in Applied Mathematics by the Spanish Society of Applied Mathematics (SEMA).